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Question

Let A and B any two independent events with 0<P(A)<1 and 0<P(B)<1. Which of the following statements are TRUE?

I. P(A occurs but B does not occur)=P(A)P(AB)
II. P(Exactly one of A and B occurs)=P(A)+P(B)P(AB)
III. P(Neither A nor B occurs)=(P(A)1)(P(B)1)
IV. P(A occurs given that B does not occur)=P(A)

A
Only I, II, III
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B
Only I, II, IV
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C
Only I, III, IV
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D
All of them
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Solution

The correct option is C Only I, III, IV
We know that if A and B are independent, then
1. A and BC are independent.
2. AC and B are independent.
3. AC and BC are independent.

I. P(A occurs but B does not occur)
=P(ABC)
=P(A)P(BC)
=P(A)(1P(B))
=P(A)P(A)P(B)
=P(A)P(AB)

II. P(Exactly one of A and B occurs)
=P((ABC) (ACB))
=P(A)P(BC)+P(AC)P(B)
=P(A)+P(B)2P(AB)

III. P(Neither A nor B occurs)
=P(ACBC)
=P(AC)P(BC)
=(1P(A))(1P(B))
=(P(A)1)(P(B)1)

IV. P(A occurs given that B does not occur)
=P(A|BC)
=P(ABC)P(BC)
=P(A)P(BC)P(BC)=P(A)
Alternatively, we can directly conclude that P(A|BC)=P(A) as A and BC are independent.

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